Fluctuations and extreme events in the public attention on Italian legislative elections

Abstract

The share of social media attention to political candidates was shown to be a good predictor of election outcomes in several studies. This attention to individual candidates fluctuates due to incoming daily news and sometimes reflects long-term trends. By analyzing Twitter data in the 2013 and 2022 election campaign we observe that, on short timescales, the dynamics can be effectively characterized by a mean-reverting diffusion process on a logarithmic scale. This implies that the response to news and the exchange of opinions on Twitter lead to attention fluctuations spanning orders of magnitudes. However, these fluctuations remain centered around certain average levels of popularity, which change slowly in contrast to the rapid daily and hourly variations driven by Twitter trends and news. In particular, on our 2013 data we are able to estimate the dominant timescale of fluctuations at around three hours. Finally, by considering the extreme data points in the tail of the attention variation distribution, we could identify critical events in the electoral campaign period and extract useful information from the flow of data.

Figure 1

Fit to the GOU process Empirical mean square log-displacement in normalized units for the four main political parties in the Italian 2013 elections, and the corresponding theoretical curves. The filtering timescale used for the instantaneous twitting rate is ${\beta }^{-1}=20$ min. We see that the GOU model, a mean-reverting diffusion process in logarithmic scale with just two parameters is able to fit the empirical curves on short timescales. GBM is Geometric Brownian motion. Precise definitions in the Methods section.

Figure 2

Comparing the distribution. Empirical distribution of the squared log-displacement in normalized units and comparison to the GOU model fit from Fig. 1. The fat tails are explained by few extreme events. Python “powerlaw” package was used for plotting. The time interval is $\tau =180$ min, larger than the filtering timescale and comparable with the mean-reversion timescale.

Figure 3

Examples of extreme events. x(t) is the share of attention at time t, and $s(t,\tau )$ is the corresponding squared log-displacement, here for an interval $\tau =30$ min and with filtering timescale ${\beta }^{-1}=20$ min. We see how significant and unexpected events in the electoral campaign are reflected in spikes of x(t).

Figure 4

Predictions using the GOU model. Time series predictions based on the GOU model with the parameters estimated immediately before prediction, at four different times during the day of 22 February 2013. The scatter plot is the realized $\tilde{x}$, while the three solid lines are the expectation and ± 2 standard deviations. The colors are as above: blue is ’PD’, orange is ’SC’, green is ’PdL’, red is ’M5S’.